Foot Locker Store Productivity. Foot Locker uses sales per...

Foot Locker Store Productivity. Foot Locker uses sales per square foot as a measure of store productivity. Sales are currently running at an annual rate of $$\$ 406$$ per square foot. You have been asked by management to conduct a study of a sample of 64 Foot Locker stores. Assume the standard deviation in annual sales per square foot for the population of all 3400 Foot Locker stores is $$\$ 80$$. a. Show the sampling distribution of $\bar{x}$, the sample mean annual sales per square foot for a sample of 64 Foot Locker stores. b. What is the probability that the sample mean will be within $$\$ 15$$ of the population mean? c. Suppose you find a sample mean of $$\$ 380$$. What is the probability of finding a sample mean of $$\$ 380$$ or less? Would you consider such a sample to be an unusually low-performing group of stores?

Foot Locker Store Productivity. Foot Locker uses sales per square foot as a measure of store productivity. Sales are currently running at an annual rate of $$\$ 406$$ per square foot. You have been asked by management to conduct a study of a sample of 64 Foot Locker stores. Assume the standard deviation in annual sales per square foot for the population of all 3400 Foot Locker stores is $$\$ 80$$.
a. Show the sampling distribution of $\bar{x}$, the sample mean annual sales per square foot for a sample of 64 Foot Locker stores.
b. What is the probability that the sample mean will be within $$\$ 15$$ of the population mean?
c. Suppose you find a sample mean of $$\$ 380$$. What is the probability of finding a sample mean of $$\$ 380$$ or less? Would you consider such a sample to be an unusually low-performing group of stores?

Key Concepts

Standard Error

The standard error is the standard deviation of the sampling distribution of the sample mean. It is calculated by dividing the population standard deviation by the square root of the sample size. This measure indicates how much variability one can expect in the sample mean from sample to sample, reflecting the precision of the sample mean as an estimate of the population mean.

Z-Score Calculation

A Z-score is a statistical measure that quantifies the number of standard deviations a data point (or in this case, a sample mean) is from the population mean. By converting a sample mean to a z-score using the standard error, one can determine the likelihood of observing such a value under the assumption of a given population mean.

Sampling Distribution of the Sample Mean

This concept refers to the distribution that would be obtained if the mean is calculated for every possible sample of a fixed size drawn from a population. It describes how the sample mean varies from sample to sample, and its distribution is centered at the population mean with a variability that decreases as the sample size increases.

Central Limit Theorem

The Central Limit Theorem states that, given a sufficiently large sample size, the distribution of the sample mean will approximate a normal distribution regardless of the shape of the underlying population distribution. This theorem is crucial because it allows for probability calculations involving the sample mean, even when the population distribution is not normal.

Assessing Unusual Outcomes

This concept involves evaluating the probability of obtaining a sample result that deviates significantly from the population parameter. When a sample mean falls far from the population mean in terms of standard errors (i.e., an extreme z-score), it may be considered unusual or indicative of a performance that is statistically unlikely, informing decisions about performance assessments.

Transcript

00:01 We'll be assuming for the footlocker that the average sales is $406 per square foot, and we have the population standard deviation to be $80 per square foot.

00:14 And they're taking a sample of size 64 from the 3 ,400 foot lockers across the nation, i guess, assumingly, it's in the united states.

00:26 And if we find that ratio, this value comes up.

00:30 To be about that's about 1 .2 1 .9 percent so this distribution we do not not need that correction factor that finite correction factor so we can go through and show what the sampling distribution of x bars is going to look like it will have a mean of x bars we'll assume is still four hundred six dollars the standard deiation of x bars would be approximately the eighty dollars divided by the square root of 64 which this is conveniently going to end up going into there it's 10 and the distribution because this sample size is sufficiently large the central limit theorem tells us that this would be approximately a normal distribution so if i just quick do a little tiny sketch here this is where the x bars would center we this would be at 416 two standard deviations for 26 go down one this would be at 396 and go down one more 386 that's within two standard deviations now part b we want to find what is the likelihood that the x bar is within the interval from the mean of 406 plus or minus 15 and so we can see that this was 10 so 15 is going to be here and up to here and so we want to find that probability and i'm going to utilize my normal cdf.

02:04 We could also convert these to z values very easily, but it looks like software is your friend here.

02:11 So if i take that 406 and subtract away, 15, we're down to 391.

02:18 And add it on, we're up to 421.

02:21 Our mean is at 406, and our standard air is at 10.

02:28 And so this will just all depends on what your class is.

02:35 This is a business class, apparently...

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